Chapter 5: Q7E (page 259)

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$

Short Answer

Expert verified

$x'_{1} (t) = x_{2} (t) x'_{2} (t) = x_{3} (t) x'_{3} (t) = x_{4} (t) + t x'_{4} (t) = x_{5} (t) x'_{5} (t) = \frac{2 x_{4} (t) - 2 x_{3} (t) + 1}{5} x_{1} (0) = x_{2} (0) = x_{3} (0) = 4, x_{4} (0) = x_{5} (0) = 1$

Step by step solution

Express equation in form of x

Here given $x''' - y = t$ and $2 x'' + 5 y'' - 2 y = 1 .$

Rewrite the equations $x''' = y + t$ and $x'' = \frac{- 5 y'' + 2 y + 1}{2}$

Denote,

$x_{1} (t) = x (t) x_{2} (t) = x' (t) x_{3} (t) = x'' (t) x_{4} (t) = y (t) x_{5} (t) = y' (t) x'_{1} (t) = x_{2} (t)$

The equation transforms as:

$x'_{1} (t) = x_{2} (t) x'_{2} (t) = x_{3} (t) x'_{3} (t) = x_{4} (t) + t x'_{4} (t) = x_{5} (t) x'_{5} (t) = \frac{2 x_{4} (t) - 2 x_{3} (t) + 1}{5}$

The initial conditions

The given initial conditions are $x (0) = x' (0) = x'' (0) = 4$ and $y (0) = y' (0) = 1 .$

Initial conditions after transformations $x_{1} (0) = x_{2} (0) = x_{3} (0) = 4, x_{4} (0) = x_{5} (0) = 1 .$

This is the required result.

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In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$

Short Answer

Step by step solution

Express equation in form of x

The initial conditions

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In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.x'''-y=t;x(0)=x'(0)=x''(0)=42x''+5y''-2y=1;y(0)=y'(0)=1

Short Answer

Step by step solution

Express equation in form of x

The initial conditions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Logic and Functions

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$